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Theorem metcnpi 21558
Description: Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 21555. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypotheses
Ref Expression
metcn.2  |-  J  =  ( MetOpen `  C )
metcn.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metcnpi  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
Distinct variable groups:    x, y, F    x, J, y    x, K, y    x, X, y   
x, Y, y    x, A, y    x, C, y   
x, D, y    x, P, y

Proof of Theorem metcnpi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr 462 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  F  e.  ( ( J  CnP  K ) `  P ) )
2 simpll 758 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  C  e.  ( *Met `  X
) )
3 simplr 760 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  D  e.  ( *Met `  Y
) )
4 eqid 2422 . . . . . . . . 9  |-  U. J  =  U. J
54cnprcl 20260 . . . . . . . 8  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  U. J )
65adantl 467 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  U. J )
7 metcn.2 . . . . . . . . 9  |-  J  =  ( MetOpen `  C )
87mopnuni 21455 . . . . . . . 8  |-  ( C  e.  ( *Met `  X )  ->  X  =  U. J )
98ad2antrr 730 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  X  =  U. J )
106, 9eleqtrrd 2510 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  X )
11 metcn.4 . . . . . . 7  |-  K  =  ( MetOpen `  D )
127, 11metcnp 21555 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z ) ) ) )
132, 3, 10, 12syl3anc 1264 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  -> 
( ( F `  P ) D ( F `  y ) )  <  z ) ) ) )
141, 13mpbid 213 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z ) ) )
1514simprd 464 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  z ) )
16 breq2 4427 . . . . . 6  |-  ( z  =  A  ->  (
( ( F `  P ) D ( F `  y ) )  <  z  <->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
1716imbi2d 317 . . . . 5  |-  ( z  =  A  ->  (
( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  z )  <-> 
( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  A ) ) )
1817rexralbidv 2944 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z )  <->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
1918rspccv 3179 . . 3  |-  ( A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
2015, 19syl 17 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
2120impr 623 1  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   U.cuni 4219   class class class wbr 4423   -->wf 5597   ` cfv 5601  (class class class)co 6306    < clt 9683   RR+crp 11310   *Metcxmt 18955   MetOpencmopn 18960    CnP ccnp 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-sup 7966  df-inf 7967  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-n0 10878  df-z 10946  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-bl 18965  df-mopn 18966  df-top 19920  df-bases 19921  df-topon 19922  df-cnp 20243
This theorem is referenced by: (None)
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